An object's position as a function of time is given by r(t)=[(1m/s^2)t^2] i−[(3m/s)t+3m)] jThe object moves without air resistance

Question

The object moves without air resistance

Part (a)  What is the object's displacement, in Cartesian unit-vector notation, between times 1s1s and 3s3s?

Part (b)  What is the object's velocity, in Cartesian unit-vector form at t=2s? Your answer should be in meters per second, but do not enter units for your equation. Your answer should be in meters, but do not enter units for your equation. 

Part (c)  What is the object's acceleration, in Cartesian unit-vector form at t=5s? Your answer should be in meters per squared second but do not enter units for your equation. 

Part (d)  What is the magnitude, in meters per second, of the object's average velocity between times 1s and 3s? 

Answer 1

[Note: units are not included in answers for simplicity: part a is in units of meters, part b in meters / second, part c in meters / second^2, and part d in meters / second]

a) The displacement is defined as r(t_f) - r(t_i) where t_f and t_i are the final and initial times respectively. We are given t_i = 0s and t_f = 3s. Substituting in we get: r(3) - r(1) = [1 * (3²) i - (3 * 3 + 3) j ] - [1 * (1²) i - (3 * 1 + 3) j] = [ 9 i - 12 j ] - [ 1 i -6 j ] = (9 - 1) i + (-12 +6) j = 8 i - 6 j

b) To find v(t), we need to take the derivative of r(t). Taking d/dt of each component we get:

v(t) = d/dt (t²) i - d/dt ( 3t + 3) j = 2t i - 3 j

Then, substituting in t = 2, we get v(2) = 2 * 2 i - 3 j = 4 i - 3 j

c) To find a(t), we need to take the second derivative of r(t), or simply the derivative of v(t). Taking d/dt of each component of v(t), we get:

a(t) = d/dt (2t) i - (d/dt) 3 j = 2 i - 0 j = 2 i (in this case there is no acceleration in the j component).

The acceleration is constant, and only in the i component, thus a(5) = 2 i

d) The average velocity is defined as v_avg = Δx / Δt or simply the displacement over the change in time. We already calculated the displacement between times 1s and 3s in part (a) as 8 i - 6 j, so we need to simply divide each component by Δt = 3s - 1s = 2s:

v_avg = (1 / 2) * 8 i - (1 / 2) * 6 j = 4 i - 3 j.

Notice this is the same answer we got for part (b). This makes sense, because the i component of v(t) changes linearly over time, and the j component is constant. So the average velocity between 1s and 3s is simply the instantaneous velocity at t = 2.

Hope this helps!